An exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained when a fluid flows in a wedge about a vortex. The map is employed to determine the potential flow outside the vortex and the vortex domain boundary provided the circulation around the vortex and constant speed on the vortex boundary are prescribed, and there are no stagnation points on the walls. The map is expressed in terms of a rational function on an elliptic surface topologically equivalent to a torus, and the solution to a symmetric Riemann–Hilbert problem on a finite and a semi-infinite segments on the same genus-1 Riemann surface. Owing to its special features, the Riemann–Hilbert problem requires a novel analogue of the Cauchy kernel on an elliptic surface. Such a kernel is proposed and employed to derive a closed-form solution to the Riemann–Hilbert problem and the associated Jacobi inversion problem. The final formula for the conformal map possesses a free geometric parameter and two model parameters. It is shown that the solution exists and the vortex has two cusps, while the solution does not exist when the wedge angle exceeds π.

当流体以楔形绕涡旋流动时，得到了从两个狭缝外部到双连通流域的共形映射的精确公式。该图用于确定涡外的势流和涡域边界，前提是规定了涡周围的环流和涡边界上的恒速，并且壁上没有停滞点。该映射表示为在拓扑上等效于环面的椭圆曲面上的有理函数，以及在同一属 1 黎曼曲面上有限和半无限段上的对称 Riemann-Hilbert 问题的解。由于其特殊的特性，黎曼-希尔伯特问题需要在椭圆表面上使用一种新的柯西核类似物。提出并使用这样的内核来导出 Riemann-Hilbert 问题和相关的 Jacobi 反演问题的封闭形式解决方案。共形图的最终公式具有一个自由几何参数和两个模型参数。证明存在解且涡旋有两个尖点，而当楔角超过π。